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Definite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDefinite Integrals5 Marks
Question
Evaluate the following definite integrals:
$\int\limits_{\frac{\pi}{2}}^{\pi}\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$
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Answer

Let $\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}$ Then,
$\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\bigg(\frac{1-2\sin\frac{\text{x}}{2}\cos\frac{\text{x}}{2}}{2\sin^2\frac{\text{x}}{2}}\bigg)\text{dx}$
$\Rightarrow\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\Big(\frac{1}{2}\text{cosec}^2\frac{\text{x}}{2}-\cot\frac{\text{x}}{2}\Big)\text{dx}$
$\Rightarrow\text{I}=\int_{\frac{\pi}{2}}^\limits{\pi}\frac{1}{2}\text{e}^{\text{x}}\text{cosec}^2\frac{\text{x}}{2}\text{ dx}-\int_{\frac{\pi}{2}}^\limits{\pi}\text{e}^{\text{x}}\cot\frac{\text{x}}{2}\text{ dx}$
Integrating second term by parts,
$\text{I}=\bigg\{-\Big[\text{e}^{\text{x}}\cot\frac{\text{x}}{2}\Big]^{\pi}_\frac{\pi}{2}-\int_{\frac{\pi}{2}}^\limits{\pi}\frac{1}{2}\text{ e}^{\text{x}}\text{cosec}^2\frac{\text{x}}{2}\text{ dx}\bigg\}+\int_{\frac{\pi}{2}}^\limits{\pi}\frac{1}{2}\text{ e}^{\text{x}}\text{cosec}^2\frac{\text{x}}{2}\text{ dx}$
$\Rightarrow\text{I}=-\Big[0-\text{e}^{\frac{\pi}{2}}\Big]$
$\Rightarrow\text{I}=\text{e}^{\frac{\pi}{2}}$

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